\section{Exclusion Limit}
\label{sec:limits}

Since no excess over the background expectation in the signal region has been observed neither for the parallel nor the perpendicular polarization, exclusion limits on the di-photon coupling strength $\gagg$ and the ALPs mass $m_a$ via Eqs.~(\ref{eq:probability}),~(\ref{eq:flux}) have been derived.

%We use a bayesian approach as described in e.g. \cite{Agashe:2014kda} to set a limit on the rate of reconverted photons $(dN/dt)$ which is used as the signal parameter using the following Likelihood model:
%\begin{equation}
    %%\mathcal{L} \propto
    %%\prod_{i} \mathcal{N}\Bigl(n_{i}^{\text{obs}} \;\Bigl| \Bigr.\;
    %%\mathcal{P}\bigl(\frac{dN}{dt} \cdot t_\text{exp}\bigr) +
    %%\mathcal{P}(\mu_{i}), \sigma_{i}\Bigr)
    %\mathcal{L} \propto \prod_i
    %\mathcal{P}(n_{\text{Obs},i}|\frac{dN}{dt}t_{\text{exp},i} + \nu_{b,i})
    %\cdot
    %\mathcal{N}(n_{\text{Obs},i}|\frac{dN}{dt}t_{\text{exp},i}+\nu_{b,i},\sigma_i)
    %\label{eq:likelihood}
%\end{equation}
%The index $i$ runs over the integer number of runs corresponding to a certain polarization state and $\mathcal{N}$ represents the Gaussian background parametrization of the $i$-th frame including an additional Poissonian signal contribution of expectation value $dN/dt$ times the frames exposure time $t_\text{exp}$. The numerical integration is done via Markov-Chain-Monte-Carlos (see e.g. \cite{Neal93probabilisticinference}) using twenty million samples for either polarization, leading to a negligible uncertainty due to the integration process.

We use a bayesian approach as described e.g.\ in \cite{Agashe:2014kda} to set limits on the rate of reconverted photons $(dN/dt)$. The likelihood model contains a poissonian contribution from the signal process and the dark count background as well as a gaussian contribution accounting for additional readout noise. All runs corresponding to a particular polarization form a combined likelihood model in which background and noise contributions are optimized according to their individual signal region sizes (c.f.\ sec.~\ref{sec:analysis}). The numerical integration is done via Markov-Chain-Monte-Carlos (see e.g. \cite{Neal93probabilisticinference}) using twenty million samples for either polarization, leading to a negligible uncertainty due to the integration process.

In order to test the stability of this approach, we imposed fake-signals with a rate of \SI{1.0e-3}{photons/s}, i.e.\ a $3\sigma$ excess w.r.t.\ our experimental sensitivity, corresponding to an hypothetical ALPs with $\gagg^{(m \rightarrow 0)} = \SI{3.9e-8}{\GeV^{-1}}$ into all frames under study to test, if the correction methodology and the limit setting procedure recovers this signal. From the posterior distribution (see Figure~\ref{fig:fake}) we infer an observed reconverted photon rate of \SI{1.1(3)e-3}{photons/s}, proving the scrutiny of the developed approach.

\begin{figure}[b]
    \centering
    \includegraphics[width=0.45\textwidth]{Figures/fake.pdf}
    \caption{Posterior distribution of reconverted photon rate with an artificially imposed fake-signal of \SI{3.0e-8}{photons/s}}
    \label{fig:fake}
\end{figure}

The $95\%$ confidence limit on the reconverted photon flux is derived via the posterior distribution of the signal parameter $dN/dt$. We obtain a limit of $\gagg<3.2\cdot10^{-8}\,\GeV^{-1}$ and $\gagg<3.5\cdot10^{-8}\,\GeV^{-1}$ for the scalar and pseudo scalar searches in the limit of a vanishing ALP mass ($m_a\rightarrow 0$). In addition, the functional relation of Eq.~(\ref{eq:probability}) is used to derive limits on $\gagg$ in dependence of the $m_a$. The results for the (pseudo-)scalar ALPSs searches are summarized in Figure~\ref{fig:limits}, where also the exclusion limits of previous experiments \cite{Ehret:2010mh}, \cite{Pugnat:2013dha}, \cite{Pugnat:2007nu} are indicated.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.45\textwidth]{Figures/exclusion_parallel.pdf}
    \includegraphics[width=0.45\textwidth]{Figures/exclusion_perpendicular.pdf}
    \caption{Exclusion limits for (pseudo-)scalar particle searches at 95\% C.L. measured by the ALPS and the OSQAR experiment.}
    \label{fig:limits}
\end{figure}

